Chow Filtration on Representation Rings of Algebraic Groups

Abstract

We introduce and study a filtration on the representation ring R(G) of an affine algebraic group G over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on R(G) and show that all three define on R(G) the same topology. For any n ≥ 1, we compute the Chow filtration on R(G) for the special orthogonal group G := O(2n + 1). In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of G over any field of characteristic ̸= 2 is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety X such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of X.